LETTERS The interaction between predation and competition Peter Chesson1 & Jessica J. Kuang1 Competition and predation are the most heavily investigated species interactions in ecology, dominating studies of species diversity maintenance. However, these two interactions are most commonly viewed highly asymmetrically. Competition for resources is seen as the primary interaction limiting diversity, with predation modifying what competition does1, although theoretical models have long supported diverse views1���5. Here we show, using a comprehensive three-trophic-level model, that competition and predation should be viewed symmetrically: these two interactions are equally able to either limit or promote diversity. Diversity maintenance requires within-species density feedback loops to be stronger than between-species feedback loops. We quantify the contributions of predation and competition to these loops in a simple, interpretable form, showing their equivalent potential to strengthen or weaken diversity maintenance. Moreover, we show that competition and predation can undermine each other, with the tendency of the stronger interaction to promote or limit diversity prevailing.Thepastfailuretoappreciatethesymmetricaleffectsand interactions of competition and predation has unduly restricted diversity maintenance studies. A multitrophic perspective should be adopted to examine a greater variety of possible effects of preda- tion than generally considered in the past. Conservation and management strategies need to be much more concerned with the implications of changes in the strengths of trophic interactions. We focus on the middle trophic level in a three-trophic-level sys- tem (Fig. 1), and address how both competition for resources (the trophic level below) and predation (the trophic level above) affect species coexistence in the middle trophic level. Several decades ago, MacArthur6 formulated the definitive model for resource competi- tion in the Lotka���Volterra form. This model leads to a measure of niche overlap, r, between any pair of species7, and also measures kj (originally kj)8 defining the fitness of any species, j. Coexistence occurs in two-species Lotka���Volterra competition if the competitive effect that a species has on the other species (interspecific competi- tion, aij) is less than the competitive effect that it has on itself (intras- pecific competition, ajj)8. Notably, the ratio of these competitive effects can be expressed in terms of fitnesses and niche overlap8: aij ajj ~ kj ki r ��1�� Species j dominates over species i, and excludes it from the system, if expression (1) is greater than one. When niche overlap is complete, r equals one and the species with the larger fitness excludes the other. Otherwise, r is less than one and the relative fitness (kj/ki) must be discounted by r (how much the species interact) to see if exclusion occurs (that is, to see if interspecific competition exceeds intraspe- cific competition). Neither species can exclude the other when expression (1) is less than one for both (i,j) 5 (1,2) and (i,j) 5 (2,1), a condition equivalent to9 rv k1 k2 v 1 r ��2�� When conditition (2) holds, the species coexist. Thus, niche overlap, r, constrains the fitness differences compatible with coexistence. Low overlap (r near to zero) means that the species can differ greatly in fitness and still coexist with each other, whereas large overlap (r near to one) means fitnesses must be nearly equal for coexistence to occur (Fig. 2). The new finding with a three-trophic-level Lotka���Volterra system is that these same conditions continue to hold, including predation in the same terms as resource competition (Box 1). To achieve this outcome, however, a new assumption is necessary: the focal species in the middle trophic level must not be the sole food source for the predators. Prey outside the focal group prevent the predators from becoming extinct when the focal species are at zero density. Although this is not the usual assumption made in models, it is not an unreas- onable case to consider: often the focal group is not the entirety of a predator���s prey, and predators often range more widely than their prey so that the predator is not solely supported by the region in which the focal group resides10. When this assumption is removed, the main conclusions here are retained (see Supplementary Information). We make a similar assumption with respect to resource competition: focal species must not drive their resources to extinction. The key conclusions are retained when this assumption is violated (Supplementary Information), but r is no longer a con- stant, complicating coexistence conditions11. In the original MacArthur model, fitness (kj) is proportional to the net excess resource intake of a species over its maintenance require- ments9. With three trophic levels, subtracted from this net excess is the mortality due to predation when predators are at their equilib- rium densities in the absence of the focal species. These new fitnesses, kj, are maximal quantities representing the abilities of focal species to gather resources and avoid predation. These quantities have the 1Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, Arizona 85721, USA. Predators Focal species Resources Figure 1 | Simplified three-trophic-level food web. The heavy lines highlight linkages between focal species through a shared resource and a shared predator. Double-headed arrows indicate that linkages are bidirectional, creating feedback loops. For example, high focal density of a species increases predator density, which then feeds back to greater predation on both the same focal species and the other focal species (apparent competition). Similarly, feedback loops through resources create resource competition. Each bidirectional linkage by itself is an intraspecific feedback loop for a focal species. Linkages between focal species through a shared predator or shared resource are interspecific feedback loops. Vol 456|13 November 2008|doi:10.1038/nature07248 235 ��2008 Macmillan Publishers Limited. All rights reserved

essential property of predicting the winning species in cases in which there is no resource or predator partitioning (that is, in situations in which there is no possibility of coexistence8,12). The quantities ajj and aij now represent the total strengths of intraspecific and interspecific density dependence, combining both competition and predation. Thus, ajj measures the combined strengths of the feedback loops from species j to species j through both resources and predators, whereas aij measures the combined strengths of all such loops from species j to species i (Fig. 1). The fact that feedback loops through predators lead to mutually negative indirect interactions between prey, analogous to competition, is the important insight of Holt2,13, who coined the term ������apparent competition������ for this outcome (Fig. 1). The simple idea that com- petitive coexistence requires intraspecific competition to exceed interspecific competition is now generalized to the idea that intra- specific density dependence must exceed interspecific density dependence. The critical ratio of interspecific density dependence to intraspecific density dependence is again given by expression (1), showing that the ability of a species to exclude another depends simply and intuitively on its relative fitness, kj/ki, discounted by niche overlap, r. Niches now involve how the species relate to their predators in addition to how they relate to their resources (Fig. 3). Niche overlap once again determines the breadth of the opportun- ities for coexistence according to condition (2), illustrated in Fig. 2. This condition is derived from the requirement that (kj /ki)r must always be less than one for coexistence (that is, ajj should always be greater than aij). The measures ajj, aij and r depend on each feedback loop according to its strength. This fact is intuitive but of profound consequence: competition and predation can each undermine the predicted effects of the other (either coexistence or exclusion) depending on which is stronger. Niche overlap jointly represents the overlap between species in their patterns of resource dependenceand their patterns of predator susceptibility (Fig. 3), but the dependence of r on predators and resources reflects the tendencies of these trophic levels to dominate focal species interactions. If resources strongly dominate, r approaches the limiting value rR based on resource overlap alone. If predators strongly dominate, r approaches the predator overlap value rP. Which of these dominates depends on the relative strengths of the density-dependent feedback loops through resources and through predators (that is, on which of these more strongly regulates the densities of the focal species). A complex of factors determine which feedback loops are strongest, but, simply put, resource loops are strong if resources regenerate slowly, and predation loops are strong ifpredatorsare primarily controlled byprey inthefocalgroup(Box 1). Whether coexistence or exclusion is promoted is determined by whether partitioning of the dominant interaction occurs���be that competition or predation. As the relative intensity of predation and competition is changed, niche overlap, r, changes as depicted in Fig. 4. Cases in which there is resource partitioning, but no predator partitioning (Fig. 3b), make r an increasing function of relative preda- tion intensity, having a low value when competition is dominant, increasing to a value of one when predation dominates (Fig. 4, curve b). Thus, broad opportunities for coexistence in terms of potentially broad differences in k values are permitted when competition domi- nates,butnotwhenpredationdominates.Notably,theoppositepattern of predator partitioning without resource partitioning (Fig. 3c) pro- vides the strongestopportunities forcoexistence (lowest r)under dom- inance by predation (Fig. 4, curve c). When there is no partitioning at either level, coexistence is still possible if there is a trade-off across species between resource sensitivity and predation sensitivity (Fig. 3d). In this case, opportunities for coexistence arise for a broad region of intermediate values of relative competition and predation intensities (Fig. 4, curve d). However, in the absence of the trade-off between competition and predation, r is instead one for all predation and competition intensities (Fig. 4, curve e). 0 0.2 0.4 r 0.6 0.8 1.0 101 100 10���1 Coexistence Species 1 dominates Species 2 dominates k 1 / k 2 Figure 2 | Coexistence and exclusion regions. Two species coexist when the niche overlap, r, and fitness ratio, k1/k2, lie within the central wedge, in which condition (2) is satisfied. Exclusion occurs outside this wedge. The log scale for the k1/k2-axis preserves symmetry. Box 1 | Model and analysis Lotka���Volterra equations for three trophic levels (focal species, Nj, resources, Rl, and predators, Pm) are 1 Nj dNj dt ~ X l cjlvlRl{ X m ajmPm{mj 1 Rl dRl dt ~rlR 1{alRRl { X j Njcjl 1 Pm dPm dt ~rm P 1{amPm P z X j wNjajm ��3�� with parameters cjl (consumption of resource l by focal species j), ajm (attack rate of focal species j by predator m), rlR and rm P (predator and resource intrinsic rates of increase), alR and am(resource P and predator intraspecific competition���reciprocals of carrying capacities), vl (unit value of resource l), mj (resource maintenance requirement of focal species j), and w (value of a unit of prey to a predator). For any pair, j and k, of focal species, methods previously described7 give the overlap measure r~ X l cjlvlckl rlRalR z X m ajmwakm rmamPP X l cjl 2vl rlRalR z X m ajmw 2 rmam P P ! X l cklvl 2 rlRalR z X m akmw2 rmamPP ! v��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� u u t ��4�� To obtain rR and rP, the predator and resource terms, respectively, are set to zero. Joint sensitivity to predation and competition is measured as sj~t X l cjl 2vl rlRalR z X m ajmw2 rmamPP v������������������������������������������������������������������������������������������������������������������������������������������������! u u ��5�� following Appendix D of ref. 9. Fitness measures, kj~ 1 sj X l cjlvl alR { X m ajm am P {mj ! ��6�� are focal species per capita growth rates at zero densities of all focal species, divided by sj (Appendix D of ref. 9). Intraspecific and interspecific coefficients of density dependence are ajj~sj=kj and aij~rsj=ki ��7�� as explained in Supplementary Information. The invasibility criterion for coexistence of two species8 leads to condition (2). See Supplementary Information for details. Competition is strong if resources regenerate slowly (that is, if rlR is small). Density dependence due to predation is strong if predators depend only weakly on prey outside the focal group (that is, if rm P is small). The importance of predation or competition is thus inverse to rlR or rm. P Figure 4 represents a common linear increase from left to right in each 1=rm P with a corresponding linear decrease in each 1=rlR. LETTERS NATURE|Vol 456|13 November 2008 236 ��2008 Macmillan Publishers Limited. All rights reserved